I have the following optimization problem \begin{array}{ll} \mathrm{maximize} & \displaystyle\sum_{i=1}^j k_i y_i,\\ \text{subject to} & \displaystyle\sum_{i=1}^j \log_{2}(1+ k_iz_i)\geq C,\\ & 0 \leq y_i+z_i \leq q,~y_{i}\geq0,~ z_{i} \geq 0,\end{array} where $k_i$, $q$ and $C$ are positive and real numbers. Both $y_i$ and $z_i$ are variables. I think we can get ride of the variable $z_{i}$ and update the problem as the following: \begin{array}{ll} \mathrm{maximize} & \displaystyle\sum_{i=1}^j k_i y_i,\\ \text{subject to} & \displaystyle\sum_{i=1}^j \log_{2}(1+ k_i(q-y_i))\geq C,\\ & 0 \leq y_i \leq q.\end{array} Therefore, the Lagrangian function will be: $$\mathcal{L}(y,\lambda,\mu,\nu)=\sum_{i=1}^{j}k_iy_i+\lambda\big(\sum_{i=1}^{j}\log_2(1+k_i(q-y_i))-C\big)+\mu_{i}(q-y_i)+\nu_{i}y_{i} $$ Assuming the problem is feasible, Using the KKT conditions and thinking about the complementary slackness: $\lambda>0$ which is true, but I think $\mu_{i}$ and $\nu_{i}$ are little bit tricky. In my opnion, if $y_i=q$, then $\mu_i>0$, and $\nu_i=0$ and if $0<y_i<q$ , then $\mu_{i}=0$ and $\nu_{i}=0$ and finally if $y_i=0$, then $\mu_i>0$ and $\nu_i>0$. Therefor, I was not able to solve the problem based on the KKT system of equations(in case my analysis was correct). Any idea if there is a closed form solution for the problem?
Thanks